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Minimal consistent EDOs: Difference between revisions

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An [[edo]] ''N'' is ''[[consistent]]'' with respect to the [[Odd limit|''q''-odd-limit]] if the closest approximations of the odd harmonics of the q-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics. It is ''[[distinctly consistent]]'' if every one of those closest approximations is a distinct value, and ''purely consistent'' if its [[relative interval error|relative errors]] on odd harmonics up to and including ''q'' never exceed 25%. Below is a table of the smallest consistent, and the smallest distinctly consistent, edo for every odd number up to 135. Odd limits of {{nowrap|2<sup>''n''</sup> &minus; 1}} are '''highlighted'''.  
An [[edo]] ''N'' is ''[[consistent]]'' with respect to the [[Odd limit|''q''-odd-limit]] if the closest approximations of the odd harmonics of the q-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics. It is ''[[distinctly consistent]]'' if every one of those closest approximations is a distinct value, and ''purely consistent'' if its [[relative interval error|relative errors]] on odd harmonics up to and including ''q'' never exceed 25%. It is accurate if every interval in that odd-limit are consistent to at least [[Harmonic distance|distance 2]], or at most 25% relative error. Below is a table of the smallest consistent, and the smallest distinctly consistent, edo for every odd number up to 135. Odd limits of {{nowrap|2<sup>''n''</sup> &minus; 1}} are '''highlighted'''.  


<onlyinclude>{| class="wikitable center-all"
<onlyinclude><includeonly>
<includeonly>
|+ style="font-size: 105%;" | Smallest consistent EDOs per odd limit
|+ style="font-size: 105%;" | Smallest consistent EDOs per odd limit
</includeonly>
</includeonly>
{| class="wikitable center-all"
|-
|-
! Odd<br />limit !! Smallest<br />consistent edo&#42; !! Smallest distinctly<br />consistent edo !! Smallest ''purely<br />consistent''&#42;&#42; edo
! Odd<br />limit !! Smallest<br />consistent edo&#42; !! Smallest distinctly<br />consistent edo !! Smallest ''purely<br />consistent''&#42;&#42; edo
!Smallest
accurate**** edo
!Smallest distinctly
accurate**** edo
|- style="font-weight: bold; background-color: #dddddd;"
|- style="font-weight: bold; background-color: #dddddd;"
| 1 || 1 || 1 || 1
| 1 || 1 || 1 || 1
|1
|1
|- style="font-weight: bold; background-color: #dddddd;"
|- style="font-weight: bold; background-color: #dddddd;"
| 3 || 1 || 3 || 2
| 3 || 1 || 3 || 2
|2
|3
|-
|-
| 5 || 3 || 9 || 3
| 5 || 3 || 9 || 3
|3
|12
|- style="font-weight: bold; background-color: #dddddd;"
|- style="font-weight: bold; background-color: #dddddd;"
| 7 || 4 || 27 || 10
| 7 || 4 || 27 || 10
|31
|31
|-
|-
| 9 || 5 || 41 || 41
| 9 || 5 || 41 || 41
|41
|41
|-
|-
| 11 || 22 || 58 || 41
| 11 || 22 || 58 || 41
|72
|72
|-
|-
| 13 || 26 || 87 || 46
| 13 || 26 || 87 || 46
|270
|270
|- style="font-weight: bold; background-color: #dddddd;"
|- style="font-weight: bold; background-color: #dddddd;"
| 15 || 29 || 111 || 87
| 15 || 29 || 111 || 87
|494
|494
|-
|-
| 17 || 58 || 149 || 311
| 17 || 58 || 149 || 311
|
|
|-
|-
| 19 || 80 || 217 || 311
| 19 || 80 || 217 || 311
|
|
|-
|-
| 21 || 94 || 282 || 311
| 21 || 94 || 282 || 311
|
|
|-
|-
| 23 || 94 || 282 || 311
| 23 || 94 || 282 || 311
|
|
|-
|-
| 25 || 282 || 388 || 311
| 25 || 282 || 388 || 311
|
|
|-
|-
| 27 || 282 || 388 || 311
| 27 || 282 || 388 || 311
|
|
|-
|-
| 29 || 282 || 1323 || 311
| 29 || 282 || 1323 || 311
|
|
|- style="font-weight: bold; background-color: #dddddd;"
|- style="font-weight: bold; background-color: #dddddd;"
| 31 || 311 || 1600 || 311
| 31 || 311 || 1600 || 311
|
|
|-
|-
| 33 || 311 || 1600 || 311
| 33 || 311 || 1600 || 311
|
|
|-
|-
| 35 || 311 || 1600 || 311
| 35 || 311 || 1600 || 311
|
|
|-
|-
| 37 || 311 || 1600 || 311
| 37 || 311 || 1600 || 311
|
|
|-
|-
| 39 || 311 || 2554 || 311
| 39 || 311 || 2554 || 311
|
|
|-
|-
| 41 || 311 || 2554 || 311
| 41 || 311 || 2554 || 311
|
|
|-
|-
| 43 || 17461 || 17461 || 20567
| 43 || 17461 || 17461 || 20567
|
|
|-
|-
| 45 || 17461 || 17461 || 20567
| 45 || 17461 || 17461 || 20567
|
|
|-
|-
| 47 || 20567 || 20567 || 20567
| 47 || 20567 || 20567 || 20567
|
|
|-
|-
| 49 || 20567 || 20567 || 459944
| 49 || 20567 || 20567 || 459944
|
|
|-
|-
| 51 || 20567 || 20567 || 459944
| 51 || 20567 || 20567 || 459944
|
|
|-
|-
| 53 || 20567 || 20567 || 1705229
| 53 || 20567 || 20567 || 1705229
|
|
|-
|-
| 55 || 20567 || 20567 || 1705229
| 55 || 20567 || 20567 || 1705229
|
|
|-
|-
| 57 || 20567 || 20567 || 1705229
| 57 || 20567 || 20567 || 1705229
|
|
|-
|-
| 59 || 253389 || 253389 || 3159811
| 59 || 253389 || 253389 || 3159811
|
|
|-
|-
| 61 || 625534 || 625534 || 3159811
| 61 || 625534 || 625534 || 3159811
|
|
|- style="font-weight: bold; background-color: #dddddd;"
|- style="font-weight: bold; background-color: #dddddd;"
| 63 || 625534 || 625534 || 3159811
| 63 || 625534 || 625534 || 3159811
|
|
|-
|-
| 65 || 625534 || 625534 || 3159811
| 65 || 625534 || 625534 || 3159811
|
|
|-
|-
| 67 || 625534 || 625534 || 7317929
| 67 || 625534 || 625534 || 7317929
|
|
|-
|-
| 69 || 759630 || 759630 || 8595351
| 69 || 759630 || 759630 || 8595351
|
|
|-
|-
| 71 || 759630 || 759630 || 8595351
| 71 || 759630 || 759630 || 8595351
|
|
|-
|-
| 73 || 759630 || 759630 || 27783092
| 73 || 759630 || 759630 || 27783092
|
|
|-
|-
| 75 || 2157429 || 2157429 || 34531581
| 75 || 2157429 || 2157429 || 34531581
|
|
|-
|-
| 77 || 2157429 || 2157429 || 34531581
| 77 || 2157429 || 2157429 || 34531581
|
|
|-
|-
| 79 || 2901533 || 2901533 || 50203972
| 79 || 2901533 || 2901533 || 50203972
|
|
|-
|-
| 81 || 2901533 || 2901533 || 50203972
| 81 || 2901533 || 2901533 || 50203972
|
|
|-
|-
| 83 || 2901533 || 2901533 || 50203972
| 83 || 2901533 || 2901533 || 50203972
|
|
|-
|-
| 85 || 2901533 || 2901533 || 50203972
| 85 || 2901533 || 2901533 || 50203972
|
|
|-
|-
| 87 || 2901533 || 2901533 || 50203972
| 87 || 2901533 || 2901533 || 50203972
|
|
|-
|-
| 89 || 2901533 || 2901533 || 50203972
| 89 || 2901533 || 2901533 || 50203972
|
|
|-
|-
| 91 || 2901533 || 2901533 || 50203972
| 91 || 2901533 || 2901533 || 50203972
|
|
|-
|-
| 93 || 2901533 || 2901533 || 50203972
| 93 || 2901533 || 2901533 || 50203972
|
|
|-
|-
| 95 || 2901533 || 2901533 || 50203972
| 95 || 2901533 || 2901533 || 50203972
|
|
|-
|-
| 97 || 2901533 || 2901533 || 1297643131
| 97 || 2901533 || 2901533 || 1297643131
|
|
|-
|-
| 99 || 2901533 || 2901533 || 1297643131
| 99 || 2901533 || 2901533 || 1297643131
|
|
|-
|-
| 101 || 2901533 || 2901533 || 3888109922
| 101 || 2901533 || 2901533 || 3888109922
|
|
|-
|-
| 103 || 2901533 || 2901533 || 3888109922
| 103 || 2901533 || 2901533 || 3888109922
|
|
|-
|-
| 105 || 2901533 || 2901533 || 3888109922
| 105 || 2901533 || 2901533 || 3888109922
|
|
|-
|-
| 107 || 2901533 || 2901533 || 13805152233
| 107 || 2901533 || 2901533 || 13805152233
|
|
|-
|-
| 109 || 2901533 || 2901533 || 27218556026
| 109 || 2901533 || 2901533 || 27218556026
|
|
|-
|-
| 111 || 2901533 || 2901533 || 27218556026
| 111 || 2901533 || 2901533 || 27218556026
|
|
|-
|-
| 113 || 2901533 || 2901533 || 27218556026
| 113 || 2901533 || 2901533 || 27218556026
|
|
|-
|-
| 115 || 2901533 || 2901533 || 27218556026
| 115 || 2901533 || 2901533 || 27218556026
|
|
|-
|-
| 117 || 2901533 || 2901533 || 27218556026
| 117 || 2901533 || 2901533 || 27218556026
|
|
|-
|-
| 119 || 2901533 || 2901533 || 42586208631
| 119 || 2901533 || 2901533 || 42586208631
|
|
|-
|-
| 121 || 2901533 || 2901533 || 42586208631
| 121 || 2901533 || 2901533 || 42586208631
|
|
|-
|-
| 123 || 2901533 || 2901533 || 42586208631
| 123 || 2901533 || 2901533 || 42586208631
|
|
|-
|-
| 125 || 2901533 || 2901533 || 42586208631
| 125 || 2901533 || 2901533 || 42586208631
|
|
|- style="font-weight: bold; background-color: #dddddd;"
|- style="font-weight: bold; background-color: #dddddd;"
| 127 || 2901533 || 2901533 || 42586208631
| 127 || 2901533 || 2901533 || 42586208631
|
|
|-
|-
| 129 || 2901533 || 2901533 || 42586208631
| 129 || 2901533 || 2901533 || 42586208631
|
|
|-
|-
| 131 || 2901533 || 2901533 || 93678217813***
| 131 || 2901533 || 2901533 || 93678217813***
|
|
|-
|-
| 133 || 70910024 || 70910024 || 93678217813
| 133 || 70910024 || 70910024 || 93678217813
|
|
|-
|-
| 135 || 70910024 || 70910024 || 93678217813
| 135 || 70910024 || 70910024 || 93678217813
{{Table notes|cols=4
 
|
|{{Table notes|cols=4
| Apart from 0edo
| Apart from 0edo
| ''Purely consistent'' is an {{idiosyncratic}}
| ''Purely consistent'' is an {{idiosyncratic}}
| Purely consistent to the 137-odd-limit
| Purely consistent to the 137-odd-limit
}}
}}
|}</onlyinclude>
|}
</onlyinclude>


The last entry, 70910024edo, is consistent up to the 135-odd-limit. The next edo is [[5407372813edo|5407372813]], reported to be consistent to the 155-odd-limit.
The last entry, 70910024edo, is consistent up to the 135-odd-limit. The next edo is [[5407372813edo|5407372813]], reported to be consistent to the 155-odd-limit.
Retrieved from "https://en.xen.wiki/w/Minimal_consistent_EDOs"